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Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 32 (0 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
Physical Hypercomputation and the Church–Turing Thesis
, 2003
"... We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a ..."
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Cited by 14 (1 self)
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We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy’s thesis.
The history and concept of computability
 in Handbook of Computability Theory
, 1999
"... We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in th ..."
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Cited by 5 (1 self)
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We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in their present roles, and how
Computability and Incomputability
"... The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal cr ..."
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Cited by 3 (0 self)
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The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alone who achieved the characterization, in the opinion of Gödel. We also explore Turing’s oracle machine and its analogous properties in analysis. Keywords: Turing amachine, computability, ChurchTuring Thesis, Kurt Gödel, Alan Turing, Turing omachine, computable approximations,
From Hypocomputation to Hypercomputation
, 2008
"... Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of ..."
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Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of metamathematics, and so many of the previous results have been folded into other theories, that we are in danger of loosing an appreciation of the broader structure of formal theories. As an aid to those looking to develop hypercomputational theories, we will briefly survey the known landmarks both inside and outside the borders of computational theory. We will not focus in this paper on why the structure of formal theory looks the way it does. Instead we will focus on what this structure looks like, moving from hypocomputational, through traditional computational theories, and then beyond to hypercomputational theories.
The Advent of Recursion . . .
"... The term ‘recursive’ has had different meanings during the past two centuries among various communities of scholars. Its historical epistemology has already been described by Soare (1996) with respect to the mathematicians, logicians, and recursivefunction theorists. The computer practitioners, on ..."
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The term ‘recursive’ has had different meanings during the past two centuries among various communities of scholars. Its historical epistemology has already been described by Soare (1996) with respect to the mathematicians, logicians, and recursivefunction theorists. The computer practitioners, on the other hand, are discussed in this paper by focusing on the definition and implementation of the ALGOL60 programming language. Recursion entered ALGOL60 in two novel ways: (i) syntactically with what we now call BNF notation, and (ii) dynamically by means of the recursive procedure. As is shown, both (i) and (ii) were introduced by linguisticallyinclined programmers who were not versed in logic and who, rather unconventionally, abstracted away from the downtoearth practicalities of their computing machines. By the end of the 1960s, some computer practitioners had become aware of the theoretical insignificance of the recursive procedure in terms of computability, though without relying on recursivefunction theory. The presented results help us to better understand the technological ancestry of modernday computer science, in the hope that contemporary researchers can more easily build upon its past.
Algorithmic Social Sciences Research Unit
, 2012
"... Bourbaki’s destructive � influence on the mathematization of economics � K. Vela Velupillai ..."
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Bourbaki’s destructive � influence on the mathematization of economics � K. Vela Velupillai
Quantum Computation: A Computer Science Perspective
, 2005
"... The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented aspects of computation and the interplay between them. This report ..."
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The theory of quantum computation is presented in a self contained way from a computer science perspective. The basics of classical computation and quantum mechanics is reviewed. The circuit model of quantum computation is presented aspects of computation and the interplay between them. This report is presented as a Master’s thesis at the department of Computer Science and Engineering at Göteborg University, Göteborg, Sweden. The text is part of a larger work that is planned to include chapters on quantum algorithms, the quantum Turing machine model and abstract approaches to quantum computation.
unknown title
"... Abstract. There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective compu ..."
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Abstract. There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy’s characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation.